Projective dimension

Remarks

One has $\operatorname{pd}_R M = 0$ if and only if $M$ is a projective module. Equivalently, $\operatorname{pd}_R M \le n$ if and only if $\operatorname{Ext}^{n+1}_R(M, N) = 0$ for every $R$-module $N$, where $\operatorname{Ext}$ is the Ext functor. Example: $\operatorname{pd}_\Z \Z/n\Z = 1$ for $n \ge 2$; the resolution $0 \to \Z \xrightarrow{\cdot n} \Z \to \Z/n\Z \to 0$ has length $1$, and $\Z/n\Z$ is not projective, because projective $\Z$-modules are free and hence have no nonzero torsion. Non-example of finiteness: over $R = \Z/4\Z$, the module $\Z/2\Z$ has $\operatorname{pd}_R \Z/2\Z = \infty$. Some authors set $\operatorname{pd}_R 0 = -\infty$.